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1.
In a previous paper of this series (Kopal, 1968a) the Eulerian equations have been set up which govern the precession and nutation of selfgravitating bodies of viscous fluid in inertial coordinates which are at rest in space. In order to facilitate their solution, in the present investigation we shall transform these equations to the rotating body-axes; and shall explicitly evaluate all their coefficients arising as a result of second-harmonic dynamical tides.Following the introductory Section 1 which contains a mathematical statement of the problem, the requisite transformation of coordinates will be outlined in Section 2, and applied to the equations of motion in Section 5. The corresponding moments and products of inertia appropriate for selfgravitating configurations of arbitrary internal structure will be formulated in Section 4; while the deformation terms arising from second-harmonic dynamical tides raised on centrally-condensed configurations will be evaluated in Sections 3 and 6. The concluding Section 7 will then contain a specification of the components of the disturbing force.The next stage of our investigation — namely, a construction of the actual solutions of the equations governing precession and nutation of fluid bodies in different cases of astrophysical interest — has been postponed for a separate paper.  相似文献   

2.
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3.
The aim of the present paper will be to give a mathematical outline of the theory of tidal evolution in close binary systems of secularly constant total momentum — an evolution activated by viscous friction of dynamical tides raised by the two components on each other. The first section contains a general outline of the problem; and in Section 2 we shall establish the basic expressions for the energy and momenta of close binaries consisting of components of arbitrary internal structure. In Section 3 we shall investigate the maximum and minimum values of the energy (kinetic and potential) which such systems can attain for given amount of total momentum; while in Section 4 we shall compare these results with the actual facts encountered in binaries with components whose internal structure (and, therefore, rotational momenta) are known to us from evidence furnished by the observed rates of apsidal advance.The results show that all such systems — be these of detached or semi-detached type — disclose that more than 99% of their total momenta are stored in the orbital momentum. The sum of the rotational momenta of the constituent components amounts to less than a percent of the total — a situation characteristic of a state close to the minimum energy for given total momentum. This appears, moreover, to be true not only of the systems with both components on the Main Sequence, but also of those possessing evolved components in contact with their Roche limits.Under such conditions, a synchronism between rotation and revolution (characteristic of both extreme states of maximum and minimum energy) is not only possible, but appears to have been actually approached — if not attained — in the majority of cases. In other words, it would appear that — in at least a large majority of known cases — the existing close binaries have already attained orbits of maximum distension consistent with their momenta; and tidal evolution alone can no longer increase the present separations of the components to any appreciable extent.The virtual absence, in the sky, of binary systems intermediate between the stages of maximum and minimum energy for given momentum leads us to conjecture that the process of dynamical evolution activated by viscous tides may enroll on a time-scale which is relatively short in comparison with their total age — even for systems like Y Cygni or AG Persei, whose total age can scarcely exceed 107 yr. A secular increase of the semi-major axes of relative orbits is dynamically coupled with a corresponding variation in the velocity of axial rotation of both components through the tidal lag arising from the viscosity of stellar material. The differential equations of so coupled a system are given in Section 5; but their solution still constitutes a task for the future.The Lunar Science Institute Contribution No. 90. The Lunar Science Institute is operated by the Universities Space Research Association under Contract No. NSR 09-051-001 with the National Aeronautics and Space Administration.  相似文献   

4.
The aim of the present paper will be to investigate the circumstances under which an irreversible dissipation of the kinetic energy into heat is generated by the dynamical tides in close binary systems if (a) their orbit is eccentric; (b) the axial rotation of the components is not synchronized with the revolution; or (c) the equatorial planes are inclined to that of the orbit.In Section 2 the explicit form of the viscous dissipation function will be set up in terms of the velocity-components of spheroidal deformation arising from the tides; in Section 3, the principal partial tides contributing to the dissipation will be detailed; Section 4 will be devoted to a determination of the extent of stellar viscosity — both gas and radiative; while in the concluding Section 5 quantitative estimates will be given of the actual rate at which the kinetic energy of dynamical tides gets dissipated into heat by viscous friction in stellar plasma.The results disclose that the amount of heat produced per unit time by tidal interaction between components of actual close binaries equals only about 10–10th part of their nuclear energy production; and cannot, therefore, affect the internal structure of evolution of the constituent stars to any appreciable extent. Moreover, it is shown that the kinetic energy of their axial rotation can be influenced by tidal friction only on a nuclear, rather than gravitational (Kelvin) time-scale — as long as plasma or radiative viscosity constitute the sole sources of dissipation. However, the emergence of turbulent viscosity in secondary components of late spectral types, which have evolved away from the Main Sequence, can accelerate the dissipation 105–106 times, and thus give rise to appreciable changes in the elements of the system (particularly, in the orbital periods) over time intervals of the order of 105–106 years. Lastly, it is pointed out that, in close binary systems consisting of a pair of white dwarfs, a dissipation of the kinetic energy through viscous tides in degenerate fermion-gas could produce enough heat to account, by itself, for the observed luminosity of such objects.  相似文献   

5.
In a preceding paper (Kopal, 1969; in what follows referred to as Paper I) we introduced a new system of curvilinear coordinates-hereafter referred to as Roche Coordinates — in which spheres of constant radius in spherical polars have been replaced by surfaces of constant potential of a rotating gravitational dipole; while the angular coordinates are orthogonal to the equipotentials. In Paper I we established an explicit form of such a transformation, and related the Roche coordinates with polar coordinates (with which they coalesce in the immediate neighbourhood of each one of the two finite mass-points) in the plane case. The aim of the present investigation will be to generalize the definition of the Roche coordinates to three dimensions.The opening Section 1 of this paper will contain a general outline of the proposed three-dimensional transformation; and in Section 2 details of this transformation will be explicitly worked out correctly to quantities of first order in superficial distortion — an approximation which should prove adequate in regions surrounding the two finite masses; while in Section 3 we shall evaluate (to this degree of accuracy) the metric coefficients of the respective transformation, and its direction cosines, in both polar and curvilinear coordinates. Section 4 will then contain a formulation of the fundamental equations of hydrodynamics in terms of the three-dimensional Roche coordinates; and their advantages for a treatment of certain classes of dynamical problems encountered in doublestar astronomy will be illustrated in the concluding Section 5 by an investigation of the vibrational stability of the Roche model. We shall show that this model is capable of performing free radial oscillations which remain barotropic only if its equilibrium form is spherical (i.e., in the absence of any external mass in the neighbourhood); but not if it is distorted to any extent by rotation or tides.  相似文献   

6.
The aim of the present paper will be to utilize the results obtained in the preceding papers of this series for the development of practical procedures for obtaining the elements of any eclipsing system from the observed photometric data by their analysis in the frequency-domain, for any type of eclipses, any proximity of the two components, and any degree of the law of limbdarkening of the eclipsed star.In Section 2, which follows a brief introduction to the subject, procedures will be developed which should permit us to perform such an analysis — by hand or automatic machine computation — for the case of mutual eclipses in binary systems the components of which can be regarded as spheres; and whose apparent discs are characterized by an arbitrary radially symmetrical distribution of surface brightness. In Section 3 we shall generalize these procedures to systems consisting of arbitrarily distorted stars.Paper dedicated to Professor Hannes Alfvén on the occasion of his 70th birthday, 30 May, 1978.  相似文献   

7.
The aim of the present paper will be to detail the explicit form of the equations which govern first-order oscillations of fast-rotating globes of self-gravitating fluids; with due account taken of the effects arising from the centrifugal as well as Coriolis force. As such configurations oscillate in general about distorted figures of equilibrium, the equations governing them can be conveniently expressed in terms of the Clairaut coordinates, associated with distorted spheroidal figures, and introduced in our previous paper (Kopal, 1980) for this purpose.In Section 2 which follows a brief outline of our problem, the equilibrium properties of fast-rotating configurations or arbitrary structure will be formulated. In Section 3 we shall carry out a separation of the variables in the equations of motion, and reduce the partial differential equations of the problem to an equivalent system of ordinary differential equations, by an expansion of expressions for the velocity componentsU, V, W in terms of tesseral harmonicsY n m (, ). The explicit form of such a system, including the effects of all tesseral harmonics of orders up tom=n=4, will be specified in Section 3 for configurations whose equilibrium form is a sphere; while in Section 4 this latter condition will be relaxed to allow for the equilibrium configuration to become a rotational spheroid.In the concluding Section 5 we shall convert the complex form of our equations of motion into real terms, amenable to a solution-analytical or numerical-in terms of real variables; and shall establish the boundary conditions necessary for a specification of the characteristic frequencies of oscillation.  相似文献   

8.
The aim of the present investigation will be to determine the explicit forms of differential equations which govern secular perturbations of the orbital elements of close binary systems in the plane of the orbit (i.e., of the semi-major axisA, eccentricitye, and longitude of the periastron ), arising from the lag of dynamical tides due to viscosity of stellar material. The results obtained are exact for any value of orbital eccentricity comprised between 0e<1; and include the effects produced by the second, third and fourth-harmonic dynamical tides, as well as by axial rotation with arbitrary inclination of the equator to the orbital plane.In Section 2 following brief introductory remarks the variational equations of the problem of plane motion will be set up in terms of the rectangular componentsR, S, W of disturbing accelerations with respect to a revolving system of coordinates. The explicit form of these coefficients will be established in Section 3 to the degree of accuracy to which squares and higher powers of quantities of the order of superficial distortion can be ignored. Section 4 will be devoted to a derivation of the explicit form of the variational equations for the case of a perturbing function arising from axial rotation; and in Section 5 we shall derive variational equations which govern the perturbation of orbital elements caused by lagging dynamical tides.Numerical integrations of these equations, which govern the tidal evolution of close binary systems prompted by viscous friction at constant mass, are being postponed for subsequent investigations.Prepared at the Lunar Science Institute, Houston, Texas, under the joint support of the Universities Space Research Association, Charlottesville, Virginia, and the National Aeronautics and Space Administration Manned Spacecraft Center, Houston, Texas, under Contract No. NSR 09-051-001. This paper constitutes Lunar Science Institute Contribution no. 100.Normally at the Department of Astronomy, University of Manchester, England.  相似文献   

9.
The aim of the present paper will be to extend the Fourier methods of analysis of the light curves of eclipsing binaries, outlined in our previous communication (Kopal, 1975) in connection with systems whose components would appear as uniformly bright discs, to systems whose components exhibit discs characterized by an arbitrary radially-symmetrical distribution of brightness —i.e., an arbitrary law of darkening towards the limb — be it linear or nonlinear.In Section 2 which follows a few brief introductory remarks, fundamental equations will be set up which govern the light changes arising from the mutual eclipses of limb-darkened stars — be such eclipses total, partial or annular; and Section 3 will contain a closed algebraic solution for the elements of the occulation eclipses terminating in total phase. Such a solution proves to be no more complicated than it turned out to be for uniformly bright discs in our previous paper; and calls for no special functions for the purpose — as will be put in proper perspective in the concluding Section 4.The cases of transit eclipses terminating in an annular phase, of partial eclipses of occulation or transit type, will be similarly dealt with by Fourier methods in the next paper of the present series.  相似文献   

10.
The aim of the present study has been to set the system of differential equations which govern the precession and nutation of self-gravitating globes of compressible viscous fluid, due to the attraction exerted on the rotating configuration by its companion; and to construct their approximate solution which are correct to terms of the second order in small dependent variables of the problem. Section 2 contains an explicit formulation of the effects of viscosity arising in this connection, given exactly as far as the viscosity remains a function of radial distancer only; but irrespective of its magnitude. In Section 3 the equations of motion will be linearized for the case of near-circular orbits and small inclinations andi of the equator of the rotating configuration, and of its orbital plane, to the invariable plane of the system; while in Section 4 further simplifications will be introduced which are legitimate for studies of secular (or long-periodic) motions of the nodes and inclinations. The actual solutions of so simplified a system of equations are constructed in Section 5; and these represent a generalization of the results obtained in our previous investigation (Kopal, 1969) of the inviscid case.The physical significance of the new results will be discussed in the concluding Section 6. It is demonstrated that the axes of rotation of deformable components in close binary systems are initially inclined to the orbital plane, viscous dissipation produced by dynamical tides will tend secularly to rectify their positions until perpendicularity to the orbital plane has been established, and the equators as well as orbit made to coincide with the invariable plane of the system-in a similar manner as other effects of tidal friction are bound eventually to synchronize the velocity of axial rotation with that of orbital revolution in the course of time.An application of the results of the present study to the dynamics of the Earth-Moon system discloses that the observed inclination of 1°.5 of the lunar equator to the ecliptic cannot be regarded as being secularly constant, but representing the present deviations from perpendicularity of oscillatory motion of very long period.The Lunar Science Institute is operated by the Universities Space Research Association under Contract No. NSR-09-051-001 with the National Aeronautics and Space Administration. This paper constitutes the Lunar Science Institute Contribution No. 85.  相似文献   

11.
In a series of papers, the equilibrium configurations of highly rotating fluid bodies have been derived. The deformation of these inhomogeneous self-gravitating fluid, of arbitrary internal structure are due to centrifugation potential. These level surfaces are expressed in terms of fourth-order sectorial harmonics.In this paper, the main equations of the problem — such as the surface of the distorted body, the gravitational potential at an arbitrary point and the disturbing potential — have been expanded to the fourth-order in terms of the even-order sectorial harmonics.This work will hereafter be referred to as Paper I.  相似文献   

12.
In continuation of our preceding investigation (Kopal, 1982c) of photometric consequences of free oscillations — radial or nonradial — of the eclipsing component on the light changes of close binary systems, the aim of the present paper will be to extend the scope of our inquiry to include the effects of similar oscillations of the component which undergoes eclipse, with arbitrary period and phase.In Sections 2 and 3 which follow a brief introduction to the particular aspect of the problem treated in this paper, a theory will be given of photometric effects caused by arbitrary harmonic oscillations of the components of eclipsing binary systems distorted by equilibrium tides, or by axial rotation with constant angular velocity. A translation of these effects into the frequency domain constitutes a problem which is solvable in a closed form only if the eclipse in question ends in totality. In any other case, the desired effects can be mathematically described only in terms of infinite series (of satisfactory asymptotic properties); and a construction of their explicit forms will be given in the Appendix.  相似文献   

13.
In preceding papers of this series (Kopal, 1968; 1969) the Eulerian equations have been set up which govern the precession and nutation of self-gravitating fluid globes of arbitrary structures in inertial coordinates (space-axes) as well as with respect to the rotating body axes; with due account being taken of the effects arising from equilibrium as well as dynamical tides.In Section 1 of the present paper, the explicit form of these equations is recapitulated for subsequent solations. Section 2 contains then a detailed discussion of the coplanar case (in which the equation of the rotating configuration and the plane of its orbit coincide with the invariable plane of the system); and small fluctuations in the angular velocity of axial rotation arising from the tidal breathing in eccentric binary systems are investigated.In Section 3, we consider the angular velocity of rotation about theZ-axis to be constant, but allow for finite inclination of the equator to the orbital plane. The differential equations governing such a problem are set up exactly in terms of the time-dependent Eulerian angles and , and their coefficients averaged over a cycle. In Section 4, these equations are linearized by the assumption that the inclinations of the equator and the orbit to the invariable plane of the system are small enough for their squares to be negligible; and the equations of motion reduced to their canonical form.The solution of these equations — giving the periods of precession and nutation of rotating components of close binary systems, as well as the rate of nodal regression which is synchronised with precession — are expressed in terms of the physical properties of the respective system and of its constituent components; while the concluding Section 6 contains a discussion of the results, in which the differences between the precession and nutation of rigid and fluid bodies are pointed out.  相似文献   

14.
The aim of the present paper will be to establish the explicit form of the equations of radiative transfer, in plane-parallel atmospheres surrounding the stars which are distorted by axial rotation or tides, in curvilinear coordinates which parallel the distorted surface; with particular attention to the circumstances under which the effects arising from limb- and gravity-darkening are multiplicative and admit of algebraic separation. In Section 2 (which follows a general outline of our problem) the fundamental equations of the radiativetransfer problem will be formulated for the ‘grey’ case; and rewritten in Section 3 in terms of non-orthogonal coordinates in which the potential over a level surface in hydrostatic equilibrium replaces the radial coordinate of spherical polars. In Section 4 we shall proceed to construct an explicit solution of the corresponding transfer problem in a plane-parallel approximation; and to prove that the effects of limb- and gravity-darkening remain factorizable only to terms which are linear in the cosines μ of the angle of foreshortening. Lastly, in Section 5 we shall list additional problems, arising in this connection, which still await appropriate treatment.  相似文献   

15.
The aim of the present paper will be to utilize Poincaré's criteria to investigate secular stability of self-gravitating configurations, of arbitrary structure, in the state of rapid rotation. The potential energy, a knowledge of which is necessary for application of these criteria, will be determined by an extension of Clairaut's method; and its evaluation in terms of suitably chosen generalized coordinates carried out explicitly to quantities of fourth order in superficial oblateness, for configurations of arbitrary internal structure.The method employed can, moreover, clearly be extended to attain accuracy of any order — at the expense of mere manipulative work which lends itself to machine automation; and the angular velocity of axial rotation can be an arbitrary function of position as well as of the time. An application of our results to homogeneous configurations in rigig-body rotation will be undertaken to demonstrate that our method, when applied to a case for which a closed solution exists, leads to results which are consistent with it.  相似文献   

16.
The aim of the present paper will be to generalize the concept of the Roche coordinates, introduced previously by the author (see Kopal, 1969, 1970, 1971) for a treatment of dynamical phenomena in close binary systems, to Clairaut's coordinates in which the Roche potential of a rotating dipole is replaced by the actual potential of configurations of finite density concentration and arbitrary structure.By virtue of an identification of the potential with the radial coordinate of our three-dimensional system, the Roche and Clairaut coordinates are both bound to be curvilinear if the star in question departs from spherical form. However, unlike Roche coordinates, the Clairaut coordinates introduced in this paper will not be required to constitute an orthogonal system; and, as a result of the freedom so preserved, their angular variables will be identified with the angles and of spherical polars.Such an adoption entails advantages and disadvantages. In the orthodox Roche system, the radial coordinate (i.e., the potential ) is given to us in a closed form; but their angular variables and must, in general, be obtained by an integration of partial differential equations constituting the orthogonality conditions. On the other hand for the Clairaut (non-orthogonal) system of coordinates no such integration is necessary — and, in fact, the angular variables can be adopted at will. However, their radial coordinate (i.e., the potential of a star of arbitrary structure and distortion) is no longer available in a closed form and must be constructed by a sequence of successive approximations — a process initiated in the 18th century by Clairaut (1743), which can be developed to any desired accuracy.As is well known, investigations of the stability of self-gravitating configurations of arbitrary internal structure must be conducted on the basis of fundamental equations of stellar hydrodynamics, which for small oscillations can be reduced to linear forms. In Section 2 the explicit form of these fundamental equations will be set up in Clairaut's coordinates and linearized in Section 3 to the case of small oscillations, while in Section 4 a critical comparison of the Clairaut and Roche coordinates will be made. However their application to rotating stars will be the subject of subsequent papers.  相似文献   

17.
The aim of the present paper will be to introduce a new definition of the loss of light suffered by mutual eclipses of the components of close binary systems: namely, as across-correlation of two apertures representing the eclipsing and eclipsed discs.The advantages of such a strategy over the more conventional (geometrical) approach are (a) greater symmetry of the respective expressions; (b) greater affinity of expressions arising from distortion with those expressing the light changes due to eclipses of spherical stars; and (c) greater freedom in dealing with the effects of particular distribution of brightness over the disc of the star undergoing eclipse (generalized limb-darkening), as well as of possible semi-transparency of the eclipsing component (Wolf-Rayet stars!). In point of fact, none of these tasks could be handled with equal ease by any other technique; nor could the corresponding loss of light be so automated by any other approach.In Section 2 which follows brief introductory remarks we shall evaluate the loss of light arising from distribution of brightness within the aperture undergoing eclipse, and appropriate opacity of the occulting disc. In Section 3 we shall take advantage of these new forms of our results to deduce a number of new properties of the eclipse functions — both algebraic and differential — which have so far escaped attention and which are of considerable practical interest. Lastly, in Section 4 we shall generalize the same concepts to the modification of the light changes caused by the departures of the respective apertures from circular forms.It will be shown that all these phenomena can be most naturally described in terms of Hankel transforms of the products of two Bessel functions with orders depending on the physical characteristics (distribution of brightness; opacity) of the two components; while the geometry of the system (i.e., the fractional radiir 1,2 of the two stars; or the inclinationi of their orbit) enter only through their arguments. Such formulation of our problem should bring a theory of the light changes of eclipsing variables in much closer contact with the adjacent parts of physical optics.  相似文献   

18.
The aim of the present paper will be to develop methods for computation of the Fourier transforms of the light curves of eclipsing variables — due to any type of eclipses — as a function of a continuous frequency variablev. For light curves which are symmetrical with respect to the conjunctions (but only then) these transforms prove to be real functions ofv, and expressible as rapidly convergent expansions in terms of the momentsA 2m+1 of the light curves of odd orders. The transforms are found to be strongly peaked in the low-frequency domain (attaining a maximum forv=0), and become numerically insignificant forv>3. This is even more true of their power spectra.The odd momentsA 2m+1 — not encountered so far in our previous papers — are shown in Section 3 of the present communication to be expressible as infinite series in terms of the even momentsA 2m well known to us from Papers I–IV; and polynomial expressions are developed for approximating them to any desired degree of accuracy. The numerical efficiency of such expressions will be tested in Section 4, by application to a practical case, with satisfactory results.Lastly, in Section 5, an appeal to the Wiener-Khinchin theorem (relating the power spectra with autocorrelation function of the light curves) and Parseval's theorem on Fourier series will enable us to extend our previous methods for a specification of quadratic moments of the light curves in terms of the linear ones.  相似文献   

19.
The aim of the present paper will be to develop a theory which should make it possible to investigate secular stability of close binary systems, consisting of tidally-distorted components of arbitrary internal structure, by a minimization of the potential energy of the system as a whole. In the second section which follows brief introductory remarks, appropriate expressions for the total potential energy of a close binary will be formulated. Section 3 will be concerned mainly with the nature of the tide-generating potential, and its effects on the shape of each star. In Section 4, the amplitudes of partial tides raised by this potential will be specified, for stars of arbitrary structure, correctly to terms of second order in superficial distortion; and in Section 5 we shall investigate the effects of interaction between rotation and tides to the same degree of approximation. The concluding Section 6 will then contain an explicit formulation of different constituents adding up to the total potential energy of the system, which can be used as a basis for its secular stability by the methods outlined already in our previous investigation (Kopal, 1973).  相似文献   

20.
The main aim of this paper will be to develop explicit form of the moments of the light curvesA 2m(r 1,r 2,i) required for the solution for the geometrical elementsr 1,2 andi of eclipsing systems exhibiting annular eclipses (Sections 2 and 3), as well as partial eclipses (Section 4).In the concluding Section 5 we shall demonstrate that — regardless of the type of eclipse and distribution of brightness on the apparent disc of the eclipsed star, or indeed of the shape of the eclipsing as well as eclipsed components — the momentsA 2m satisfy certain simple functional equations — a fact which relates them to other classes of functions previously studied in applied mathematics.  相似文献   

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